Mathlib v4.30 source proposition Nat.size_eq_zero

Identifier: F:ml430-nat-size-eq-zero-020ad98c
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.size_eq_zero` in the pinned Mathlib v4.30 source.

Formal statement

∀ {n : ℕ}, n.size = 0 ↔ n = 0

Dependencies

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5 direct dependencies 0 direct dependents

Evidence

kernel-Nat.size_eq_zero

Kind: kernel-term
Status: checked

Supports: ∀ {n : ℕ}, n.size = 0 ↔ n = 0

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- size_eq_zero 2>/dev/null | grep -Ec '^Nat\.size_eq_zero[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits `Nat.size_eq_zero` through the trusted `Kernel::add_declaration` gate, declared in `nat_prelude/size_extra.rs` (a new module, kept separate from `binary.rs`'s own dense `Nat.size` definition-plus-boundary set). `nat_theorem_inventory`'s rendered type is `((x0 : AxNat) -> Iff (Eq.{1} AxNat (AxNat.size x0) AxNat.zero) (Eq.{1} AxNat x0 AxNat.zero))`, matching this fact's `formal.statement`. `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. Verified both ways: the real name greps to a count `-ge 1`; grepping a made-up name (`Nat.size_eq_zero_bogus`) greps to `0`.

footprint-Nat.size_eq_zero

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by []. `nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free` additionally checks this theorem's own `Kernel::axiom_footprint` directly via `theorem_names` (this declaration's `p.size_eq_zero` entry was added there in the same session), and `the_build_is_deterministic`/`every_promised_name_is_admitted_with_the_expected_kind` (both re-run on every `nat_prelude::` sweep) confirm the declaration is admitted at the exact rendered type checked above. `nat_prelude/size_extra_tests.rs` independently checks this declaration's `axiom_footprint` too.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.size_eq_zero` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.size_eq_zero`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    }
  ]
}